"A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one." – Paul Halmos

Archive for the ‘math.AT’ Category

The problem of finding solutions to Diophantine equations can be recast in the following abstract form. Let be a commutative ring, which in the most classical case might be a number field like or the ring of integers in a number field like . Suppose we want to find solutions, over , of a system of polynomial equations

.

Then it’s not hard to see that this problem is equivalent to the problem of finding -algebra homomorphisms from to . This is equivalent to the problem of finding left inverses to the morphism

of commutative rings making an -algebra, or more geometrically equivalent to the problem of finding right inverses, or sections, of the corresponding map

of affine schemes. Allowing to be a more general scheme over can also capture more general Diophantine problems.

The problem of finding sections of a morphism – call it the section problem – is a problem that can be stated in any category, and the goal of this post is to say some things about the corresponding problem for spaces. That is, rather than try to find sections of a map between affine schemes, we’ll try to find sections of a map between spaces; this amounts, very roughly speaking, to solving a “topological Diophantine equation.” The notation here is meant to evoke a particularly interesting special case, namely that of fiber bundles.

We’ll try to justify the section problem for spaces both as an interesting problem in and of itself, capable of encoding many other nontrivial problems in topology, and as a possible source of intuition about Diophantine equations. In particular we’ll discuss what might qualify as topological analogues of the Hasse principle and the Brauer-Manin obstruction.

Let be a commutative ring. From we can construct the category of -modules, which becomes a symmetric monoidal category when equipped with the tensor product of -modules. Now, whenever we have a monoidal operation (for example, the multiplication on a ring), it’s interesting to look at the invertible things with respect to that operation (for example, the group of units of a ring). This suggests the following definition.

Definition: The Picard group of is the group of isomorphism classes of -modules which are invertible with respect to the tensor product.

By invertible we mean the following: for there exists some such that the tensor product is isomorphic to the identity for the tensor product, namely .

In this post we’ll meander through some facts about this Picard group as well as several variants, all of which capture various notions of line bundle on various kinds of spaces (where the above definition captures the notion of a line bundle on the affine scheme ).

Let be a closed orientable surface of genus . (Below we will occasionally write , omitting the genus.) Then its Euler characteristic is even. In this post we will give five proofs of this fact that do not use the fact that we can directly compute the Euler characteristic to be , roughly in increasing order of sophistication. Along the way we’ll end up encountering or proving more general results that have other interesting applications.

In this post we’ll compute the (topological) cohomology of smooth projective (complex) hypersurfaces in . When the resulting complex surfaces give nice examples of 4-manifolds, and we’ll make use of various facts about 4-manifold topology to try to say more in this case; in particular we’ll be able to compute, in a fairly indirect way, the ring structure on cohomology. This answers a question raised by Akhil Mathew in this blog post.

I passed my qualifying exam last Friday. Here is a copy of the syllabus and a transcript.

Although I’m sure there are more, I’m only aware of two other students at Berkeley who’ve posted transcripts of their quals, namely Christopher Wong and Eric Peterson. It would be nice if more people did this.

The goal of this post is to compute the cohomology of the -torus in as many ways as I can think of. Below, if no coefficient ring is specified then the coefficient ring is by default. At the end we will interpret this computation in terms of cohomology operations.

Often in mathematics we define constructions outputting objects which a priori have a certain amount of structure but which end up having more structure than is immediately obvious. For example:

Given a Lie group , its tangent space at the identity is a priori a vector space, but it ends up having the structure of a Lie algebra.

Given a space , its cohomology is a priori a graded abelian group, but it ends up having the structure of a graded ring.

Given a space , its cohomology over is a priori a graded abelian group (or a graded ring, once you make the above discovery), but it ends up having the structure of a module over the mod-Steenrod algebra.

The following question suggests itself: given a construction which we believe to output objects having a certain amount of structure, can we show that in some sense there is no extra structure to be found? For example, can we rule out the possibility that the tangent space to the identity of a Lie group has some mysterious natural trilinear operation that cannot be built out of the Lie bracket?

In this post we will answer this question for the homotopy groups of a space: that is, we will show that, in a suitable sense, each individual homotopy group is “only a group” and does not carry any additional structure. (This is not true about the collection of homotopy groups considered together: there are additional operations here like the Whitehead product.)